Expanding the Class of Free Fermions via Twin-Collapse Methods

Abstract

We present a novel graph-theoretic approach to simplifying generic many-body Hamiltonians. Our primary result introduces a recursive twin-collapse algorithm, leveraging the identification and elimination of symmetric vertex pairs (twins), as well as line-graph modules, within the frustration graph of the Hamiltonian. This method systematically block-diagonalizes Hamiltonians, significantly reducing complexity while preserving the energetic spectrum. Importantly, our approach expands the class of models that can be mapped to non-interacting fermionic Hamiltonians (free-fermion solutions), thereby broadening the applicability of classical solvability methods. Through numerical simulations on spin Hamiltonians arranged in periodic lattice configurations and Majorana Hamiltonians, we demonstrate that the twin-collapse increases the identification of simplicial and claw-free graph structures, which characterize free-fermion solvability. Finally, we extend our framework by presenting a generalized discrete Stone-von Neumann theorem. This comprehensive framework provides new insights into Hamiltonian simplification techniques, free-fermion solutions, and group-theoretical characterizations relevant for quantum chemistry, condensed matter physics, and quantum computation.

Figures (12)

• Figure 1: Example of a graph that requires multiple alternating rounds of false and true twins collapses. The graph can be fully collapse onto a single vertex by collapsing the following false and true twins in that order: $\lbrace*\rbrace{1, 2} \mapsto \lbrace*\rbrace{2}$, $\lbrace*\rbrace{2, 3} \mapsto \lbrace*\rbrace{3}$, $\lbrace*\rbrace{3, 4} \mapsto \lbrace*\rbrace{4}$, $\lbrace*\rbrace{4, 5} \mapsto \lbrace*\rbrace{5}$, $\lbrace*\rbrace{5, 6} \mapsto \lbrace*\rbrace{6}$, $\lbrace*\rbrace{6, 7} \mapsto \lbrace*\rbrace{7}$, $\lbrace*\rbrace{7, 8} \mapsto \lbrace*\rbrace{8}$, $\lbrace*\rbrace{8, 9} \mapsto \lbrace*\rbrace{9}$. Note that this order is strict since, for example, $\{3, 4\}$ is not a twin of any kind until $\{1, 2, 3\}$ have been merged.
• Figure 2: Example of a graph that is only simplicial after removing a twin vertex. Up to labelling, the non-empty cliques are $\lbrace*\rbrace{0}, \lbrace*\rbrace{1}, \lbrace*\rbrace{1, 2}, \lbrace*\rbrace{0, 1}, \lbrace*\rbrace{0, 1, 2}$, which are not simplicial. However, by removing, the vertex $6$, which is a false twin of $0$, the graph has, for example, the simplicial clique $\lbrace*\rbrace{1, 2}$.
• Figure 3: Probability, $p_{\mathrm{SCF}}$, that the $G_\mathrm{np}(n, p)$ Hamiltonians are scf, and the effect of the block diagonalisation. The $x$-axis is the edge probability $p$. In the upper plot, the thicker dashed line, $p_{\mathrm{SCF}}$, shows the probability that the according Hamiltonian is scf after the block diagonalisation. The thinner dashed line, $p_{\mathrm{SCF}}^<$, is the analytical upper bound on $p_{\mathrm{SCF}}$ before the block diagonalisation. In the lower plot, the dotted line, $\Delta \Xi$, shows how many vertices have been removed by the block diagonalisation (independent vertices in the frustration graph were ignored). The solid line, $\Delta p_{\mathrm{SCF}}$, shows the difference between the number of Hamiltonians that are scf before and after the block diagonalisation.
• Figure 4: Two-dimensional periodic brick lattice modelling the Hamiltonians. The spins are located on the vertices and we randomly draw 2-local Pauli interactions between neighboured spins. Each of the nine Pauli interactions is accepted with a certain probability $p \in (0, 1]$; the probability is the same for all edges, and we enforce that each edge has at least one non-trivial interaction. We draw the interactions separately on the edges between the vertices $1$ to $5$ and then extend the lattice periodically with periodic boundary conditions.
• Figure 5: scf probability for Hamiltonians on a periodic brick lattice. The $x$-axis is the probability $p$ that a Pauli interaction is accepted on an edge (cf. \ref{['f"brick_lattice']}). $p_{\mathrm{SCF}}$, $\Delta p_{\mathrm{SCF}}$ and $\Delta \Xi$ are as in \ref{['f"gnp']}. The dash-dotted lines are exact results for $p_{\mathrm{SCF}}$ and $\Delta p_{\mathrm{SCF}}$, respectively (cf. \ref{['s"brick_exact']}). For higher densities than the ones shown here, all plots are $0$ or close to $0$.

Theorems & Definitions (76)

• Definition 1: Modules
• Definition 2: Frustration graph
• Theorem 1: Informal
• Corollary 1
• Lemma 1: chapman_solvable_spin_models
• Proposition 1: chapman_solvable_spin_models
• Lemma 2
• Proposition 2
• Definition 3: Twin Collapse
• Corollary 2